课程名称︰回归分析
课程性质︰应数所数统组必修
课程教师︰郑明燕
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/11/7
考试时限（分钟）：15:30~18:20
试题 :
1. Consider the simple linear regression model
y =β+βx +ε,
i 0 1 i1 i
2
with E(ε)=0,Var(ε)=σ, i=1,...,n,and ε's uncorrelated.
i i i
(a)(15 pts) Suppose β is known. Find the least squares estimator of β and
0 1
find the bias and variance of this estimator.
^ ^ ^
(b)(15 pts) Suppose we have fit the simple linear regression model y=β+βx
0 1 1
with β and β both unknown, but the true regression funciton is
0 1
E(y)=β+βx +βx .
0 1 1 2 2
^ ^
Find the bias of β. Compare the variance of β with the variance of the
1 1
least squares estimator of β under the true model.
1
2. Consider two independent sets of observations
y =β+βx +ε , i=1,...,n,
1i 0 1 1i 1i
2
where the ε 's are iid Normal(0,σ ), and
1i
y =β+βx +ε , i=1,...,n,
2i 0 2 2i 2i
2
where the ε 's are iid Normal(0,σ ).Here, y 's and y 's are observations
2i 1i 2i
of the same response variable and x 's and x 's are observations of the
1i 2i
same covariate.
(a)(5 pts) Find the least squares estimator of β based on the first set of
0
observations (x ,y ),...,(x ,y ).
11 11 1n 1n
T
(b)(5 pts) Find the least squares estimator of β=(β,β,β) based on the
0 1 2
2n observations, and find its mean vector and covariance matrix.
(c)(10 pts) Compare biases and variances of the two estimators of β in (a)
0
and (b).
2
(d)(5 pts) Find an estimator for σ .
(e)(10 pts) Derive a level-α test for the hypotheses H :β=β versus
0 1 2
H :β≠β.
1 1 2
3. Consider the multiple linear regression model
y =β+βx +...+βx +ε, i=1,...,n,
i 0 1 i1 k ik i
where the ε's are uncorrelated each with zero mean and constant variance
i
2 T T T T
σ . Let β=(β,β,...,β) , y=(y ,...,y ) , ε=(ε,...,ε) and X=(x ,...x )
0 1 k 1 n 1 n 1 n
T
with x =(1,x ,...,x ) , i=1,...,n. Then the above model can be written as
i i1 ik
y=Xβ+ε.
Assume that the design matrix X is of rank k+1.
(a)(5 pts) Write down the least squares problem for estimation of β and
^
find the solution, denoted by β.
2 ^ T ^ 2
(b)(10 pts) Let s =(y-Xβ) (y-Xβ)/(n-k-1). Show that s is an unbiased
2
estimator of σ .
2 ^
(c)(10) Let r ^ be the correlation coefficient between the y 's and y 's.
yy i i
2 2
Show that r ^ = R .
yy
(d)(5 pts) Let H=(h ) be the hat matrix. Show that 0≦h ≦1, i=1,...,n.
ij ii
^
(e)(10) Show that, for any constant (k+1)-vector l, l'β is a best linear
unbiased estimator of l'β under the Gauss-Markov conditions.
4. Consider the reparameterization Y=Wα+ε of the model Y=Xβ+ε, where W=XC
and C is nonsingular.
(a)(5) Show that the hat matrices H and H for the two medels are the same.
W X
^ ^
(b)(5) Suppose α and β are least squares estimators of α and β. Express
^ ^
α in terms of β.